I currently have my US Visa in an old expired passport and a new passport. I want to renew my passport and then my US Visa shortly after, so that the visa is in the new passport and their new expiration dates become close. I have a few years left both in the visa and the passport.
While from my research there's no official restriction in doing so, are there any drawbacks I should keep in mind? For example, will this "raise a flag" in the visa renewal interview or at immigration?
No, it's fine. Since your visa application will be for the new passport, the validity/expiration of the previous one is irrelevant, and since your old US visa is close to expiry, it's perfectly normal to renew that too.
The one obvious reason most people don't do this is that renewing a passport long before expiry is effectively throwing away money, but these days this expense seems trivial compared to the massive cost and hassle of applying for US visas. (I know people who have spent thousands on travel and accommodation to apply at embassies with slightly less insane queues.)
The one issue you may run into is that some countries don't let you renew your passport until one year or so before expiry. Of course, you can always work around this by losing or destroying your passport (oops, went into the washing machine!), but then you're stuck without a passport for a while.
